Problema Solution
Problem Description A proposal before the Montana Board of Education has specified certain designations for elementary schools depending on how their students do on the Montana Evaluation of Scholastic Scoring. This test is given to all students in all public schools in grades 2 through 4. Schools that score in the top 20% are labelled excellent. Schools in the bottom 25% are labelled "in danger" and schools in the bottom 5% are designated as failing. Previous data suggests that the mean for all schools that take the Montana Evaluation of Scholastic Scoring is 75 with a standard deviation of 5.
What is the largest score that a school could get to still be considered in danger?
Answer provided by our tutors
Since we are not given any number of schools, we assume that there are 30 or more and thus the Normal Distribution would be the best choice to work with. (And the easiest)
Assuming "in danger" and "failing" are mutually exclusive groups (ie - no overlap). The the Normal Distribution is broken down into 4 categories with the following percentages:
Failing (0% - 5%), In Danger (5% - 30%), Ok (30% - 80%). amd Excellent (80% - 100%)
So we must find what score, say X, that yields a Normal Distribution Z-Score of 0.3.
Looking at the normal table (http://www.stat.ucla.edu/~ywu/teaching/normal.pdf) for a value near 0.3, we see -0.52 and -0.53. Lets choose Z(0.3) = -.0525, something in between.
Thus (X - mean)/stdev = Z(0.3) and solve for X. By substitution: (X - 75)/ 5 = -0.525
X = 75 + 5(-0.525) = 72.375
Thus X = 72.375 would be the highest score to be considered "In Danger"